Invariant manifolds and the calculation of the long-term asymptotic response of nonlinear processes using singular PDEs

Abstract

The present research work proposes a new systematic approach to the problem of finding invariant manifolds and computing the long-term asymptotic behavior of nonlinear dynamical systems. In particular, nonlinear processes are considered whose dynamics is driven by an external time-varying ‘forcing’ or input/disturbance term, or by a set of time-varying process parameters, or by the autonomous dynamics of an upstream process. The formulation of the problem is conveniently realized through a system of singular first-order quasi-linear PDEs, and a rather general set of conditions for solvability is derived. In particular, within the class of analytic solutions the aforementioned set of conditions guarantees the existence and uniqueness of a locally analytic solution. The solution of the system of singular PDEs is then proven to be a locally analytic invariant manifold of the nonlinear dynamical system. The local analyticity property of the invariant manifold enables the development of a series solution method, which can be easily implemented with the aid of a symbolic software package such as maple. Under a certain set of conditions, it is shown that the invariant manifold computed exponentially attracts all system trajectories, and therefore, the long-term asymptotic process response is described and calculated through the restriction of the process dynamics on the invariant manifold. Finally, in order to illustrate the proposed approach and method, a representative enzymatic bioreactor example is considered.